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 A234253 a(n) = sum_{i=1..n} C(7+i,8)^2. 3
 1, 82, 2107, 29332, 274357, 1930726, 10948735, 52357960, 217994860, 808970960, 2723733524, 8436372248, 24304813148, 65712993248, 167965846148, 408373664744, 949291256585, 2119095737210, 4559798912835, 9488531918460, 19148848609485, 37571357310510 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In general we have the following formula : a(n) = Sum_{i=1..n}C(e-1+i,e)^2 = C(n+e-1,e+1)*Fe(n)/C(2*e+1). We have the following definition : Fe(n) = Sum_{i=1..n}(-1)^(e+i)*C(e+i,i)*C(n+e,i), and Fe(1) = C(2*e+1,e). [This needs clarification, Joerg Arndt, May 04 2014] LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (18, -153, 816, -3060, 8568, -18564, 31824, -43758, 48620, -43758, 31824, -18564, 8568, -3060, 816, -153, 18, -1). FORMULA a(n) = sum_{i=1..n)C(7+i,8)^2 = C(n+8,9)*F8(n)/C(17,8) ; F8(n)= Sum_{i=1..n}(-1)^(8+i)*C(8+i,i)*C(n+8,i) = C(8,0)*C(n+8,0) - C(9,1)*C(n+8,1) + C(10,2)*C(n+8,2) - C(11,3)*C(n+8,3) + C(12,4)*C(n+8,4) - C(13,5)*C(n+8,5) + C(14,6)*C(n+8,6) - C(15,7)*C(n+8,7) + C(16,8)*Cn+8,8). We have the following values for F8(n) : F8(0)=1, F8(1)=24310, F8(2)=199342, F8(3)=931294, .... [This needs clarification, Joerg Arndt, May 04 2014] G.f.: x*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1) / (x-1)^18. - Colin Barker, May 02 2014 EXAMPLE For n=3, Sum_{i=1..3)C(7+i,8)^2 = C(11,9)*F8(3)/C(17,8) = 55*931294/24310 = 2107. [This needs clarification, Joerg Arndt, May 04 2014] MAPLE A234253:=n->add(binomial(7+i, 8)^2, i=1..n); seq(A234253(n), n=1..30); # Wesley Ivan Hurt, Dec 23 2013 MATHEMATICA Table[Sum[Binomial[7 + i, 8]^2, {i, n}], {n, 30}] (* Wesley Ivan Hurt, Dec 23 2013 *) CoefficientList[Series[(x^8 + 64 x^7 + 784 x^6 + 3136 x^5 + 4900 x^4 + 3136 x^3 + 784 x^2 + 64 x + 1)/(x - 1)^18, {x, 0, 40}], x] (* Vincenzo Librandi, May 06 2014 *) PROG (PARI) Vec(x*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1)/(x-1)^18 + O(x^100)) \\ Colin Barker, May 02 2014 CROSSREFS Cf. A087127, A086023, A086025, A086027, A086029. Sequence in context: A282155 A280857 A281603 * A031606 A232904 A230395 Adjacent sequences: A234250 A234251 A234252 * A234254 A234255 A234256 KEYWORD nonn,easy AUTHOR Yahia Kahloune, Dec 22 2013 EXTENSIONS One term corrected and more terms from Colin Barker, May 02 2014 STATUS approved

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Last modified June 14 04:49 EDT 2024. Contains 373393 sequences. (Running on oeis4.)